The heat equation5

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1、CHAPTER4.THEHEATEQUATION.Inthischapterweshallconsidersemi-discreteapproximationstotheinitialvalueproblemfortheheatequation.Inparticular,weshallinvestigatehowtherateofconvergenceoftheapproximatesolutiontotheexactsolutiondependsonthesmoothnessoftheinitialfunction.Theresultsareex

2、pressedintermsofBesovspacesandtheproofsusethetechniquesdevelopedinChaptersIand2.Sinceourmethodsdonotdependstronglyonthenumberofspacevariablesweconsider(forsimplicity)onlytheone-dimensionalproblem.InSectionIweproveconvergenceestimatesinLwiththesmoothnessofthePdatameasuredinthes

3、ameLspace.WealsoestimatetherateofconvergenceofPdifferencequotientsoftheapproximatesolutiontoderivativesofthesolutionofthecontinuousproblem.InSection2wederivevariousinverseresults.OntheonehandtheseshowthattheconvergenceestimatesofSectionIareinacertainsensebestpossible,andontheo

4、therhandtheymotivateanothertypeofconvergenceesti-mates,presentedinSection3,inwhichtheerrorismeasuredinthemaximumnormbutthesmoothnessofthedatainLI.Finally,inSection4weconsidertheeffectofapreliminarysmoothingoftheinitialdata.4.1.ConvergenceestimatesinL.PWeshallconsidertheinitial

5、valueproblemfortheone-dimensionalheatequa-tion~_~u=82uforxER,t>0,$t8x2"(1.1)u(x,0)=v(x).69RecallfromSection3.1thatthesolutionoperatorofthisproblemisdefinedbyE(t)v=~-1(exp(-t~2)v),andthattheproblemiswellposedinLforI

6、einitialfunctionisnot,asthefollowingresultshows:Theorem1.1.LetI0beaninteger.ThenwithE(t)aSabovethereexistsaconstantCsuchthatforv6Lp,llDaE(t)v]~~Ct-a/211Vllp,fort>0.Proof.WemaywriteD~E(t)v=~-1((i~)~exp(-t~2)v).ByTheorem1.2.8wehavefort>0,M((i~)aexp(-t~2))=t-a/2M(~exp(-

7、~2)),PPandsince~aexp(-~2)6ScMforI

8、reteproblem(3.2.5)canthenbewrittenEh(t)v=~-1(exp(-tPh)V)=~-1(